The elliptical vortices, integrable Ermakov structure, Schrödinger connection, and Lax pair in the compressible Navier–Stokes equation

• Published in: Studies in applied mathematics (Cambridge) Vol. 149; no. 4; pp. 879 - 903
• Main Authors: An, Hongli, Zhu, Haixing
• Format: Journal Article
• Language: English
• Published: Cambridge Blackwell Publishing Ltd 01-11-2022
• Subjects: Compressibility; Elliptic vortex ansatz; Ermakov systems; Fluid flow; Lax pair; Navier-Stokes equations; Nonlinear Schrödinger connection; Pulsrodon solution
• ISSN: 0022-2526; 1467-9590
• DOI: 10.1111/sapm.12524

In this paper, we investigate the (2+1)‐dimensional compressible Navier–Stokes equation with density‐dependent viscosity coefficients. We introduce a novel power‐type elliptic vortex ansatz and thereby obtain a finite‐dimensional nonlinear dynamical system. The latter is shown to not only have an underlying integrable Ermakov structure of Hamiltonian type, but also admit a Lax pair formulation and associated stationary nonlinear Schrödinger connection. In addition, we construct a class of elliptical vortex solutions termed pulsrodons corresponding to pulsating elliptic warm core eddies and discuss their dynamical behaviors. These solutions have recently found applications in geography, and oceanic and atmospheric dynamics.